\section{1D Elements}
   Spring elements include the CELAS1, CELAS2, CELAS3, CELAS4.

   The spring constant $k$ is defined for a spring.
   \subsection{Spring Element}
     \[ [K_e] = 
            \left[ \begin{array}{cc}
               k & -k \\
              -k &  k 
            \end{array}\right] \]

     \[ \left\{ \begin{array}{c}
              X_1 \\ X_2 \end{array}\right\} =
        [K_e] \left\{ \begin{array}{c}
              u_{x1} \\ u_{x2} \end{array}\right\} \]
   \subsubsection{Missing Grid Points}
   In the case of a grounded node (see the CELAS4), where the nodeID is 0,
     $[K_e]$ becomes:
     \[ [K_e] = [k]  \]
     \[ \left\{ \begin{array}{c}
              X_1 \end{array}\right\} =
        [K_e] \left\{ \begin{array}{c}
              u_{x1} \end{array}\right\} \]

   \subsection{Rod Element}
     Rod elements include the CROD, CONROD, and CTUBE. \\
     The rod element is the same as the spring, except k must be calculated:
       \[ k = \frac{AE}{L} \]\
     The same elemental stiffness matrix as the Spring Element is used:
       \[ [K_e] = 
            \left[ \begin{array}{cc}
               k & -k \\
              -k &  k 
            \end{array}\right]\]
     such that:
       \[ \left\{ \begin{array}{c}
              X_1 \\ X_2 \end{array}\right\} =
        [K_e] \left\{ \begin{array}{c}  
              u_{x1} \\ u_{x2} \end{array}\right\} \]
       
       %verify
     An extra step must be taken for the Stress/Stress Recovery step on the Rod Element.
       \[   \epsilon_{xx} = u_{x2}-u_{x1}           \]
       \[ 2 \epsilon_{xy} = \theta_{x2}-\theta_{x1} \]

       \[ \sigma_{xx} = \epsilon_{xx} E \]
       \[ \sigma_{xy} = \epsilon_{xy} G \]

   \subsection{Bar Element}
     Bar elements include the CBAR.

   \subsection{Beam Element}
     Beam elements include the CBEAM.

